Linear Spaces

Subspaces

Definition: Let VV be a vector space. A subset WW of VV is called a subspace of VV iff WW is a vector space with respect to the operations of VV, that is,

Remark: WW is a subspace of VV iff WW is nonempty and WW is closed under addition and scalar multiplication. All other axioms are inherited from the original vector space VV.


Example: linear space V=R2V = \mathbb{R}^2, subspace W=[α 0]T∈R2:α∈RW = [\alpha \space 0]^T \in \mathbb{R}^2 : \alpha \in \mathbb{R}
Solution: Let w1=[α1 0]Tw_1 = [\alpha_1 \space 0]^T and w2=[α2 0]Tw_2 = [\alpha_2 \space 0]^T and c∈Rc \in \mathbb{R},


Example: linear space V=R2V = \mathbb{R}^2, subspace W=[α 1]T∈R2:α∈RW = [\alpha \space 1]^T \in \mathbb{R}^2 : \alpha \in \mathbb{R}
Solution: Let w1=[α1 1]Tw_1 = [\alpha_1 \space 1]^T and w2=[α2 1]Tw_2 = [\alpha_2 \space 1]^T


Remark: In R2\mathbb{R}^2, a subspace is a line through the origin. Any line through the origin is a subspace of R2\mathbb{R}^2.

In function spaces, an example can be given as follows:
Example: linear space V=V = set of all real-valued functions of a real variable t→f(t)t\rightarrow f(t);
subspace W1=W_1 = set of all continuous functions [+]
subspace W2=W_2 = set of all constant functions [+]
subspace W3=W_3 = set of all functions periodic with π\pi [+] subspace W4=W_4 = set of all functions which are discontinuous at t=1t=1 [-]

Remark: 0 vector is a subspace of any vector space, even subspace of itself, and it is the smallest subspace.

[+] W1,W2,W3W_1, W_2, W_3 are subspaces of VV
W4W_4 is not a subspace of VV


Example: Show that Y+ZY+Z is a linear subspace of XX, if YY and ZZ are also linear subspaces of XX.
Proof: Let w1+w2∈Ww_1+w_2 \in W, with
w1=y1+z1w_1=y_1+z_1 where y1∈Y, z1∈Zy_1 \in Y, \ z_1 \in Z
w2=y2+z2w_2=y_2+z_2 where y2∈Y, z2∈Zy_2 \in Y, \ z_2 \in Z
then

w1+w2=y1+z1+y2+z2=(y1+y2)+(z1+z2)y1+y2∈Y,z1+z2∈Z⇒w1+w2∈Y+Z\begin{align*} w_1+w_2=y_1+z_1+y_2+z_2 &=(y_1+y_2)+(z_1+z_2) \\ y_1+y_2 \in Y , z_1+z_2 \in Z &\Rightarrow w_1+w_2 \in Y+Z \end{align*}

Shows that Y+ZY+Z is closed under addition. (lemma 1)

Let cw1∈Wc w_1 \in W, ∀c∈F\forall c \in F

cw1=c(y1+z1)=(cy1)+(cz1)cy1∈Y,cz1∈Z⇒cw1∈Y+Z\begin{align*} c w_1=c(y_1+z_1) &=(cy_1)+(cz_1) \\ cy_1 \in Y , cz_1 \in Z &\Rightarrow cw_1 \in Y+Z \end{align*}

Shows that Y+ZY+Z is closed under scalar multiplication. (lemma 2) Hence Y+ZY+Z is a linear subspace of XX.


Example: If YY and ZZ are subspaces of X, then Y∩ZY \cap Z is a subspace of XX.
 ~Proof:

now we need to show that u+w∈Y∩Zu+w \in Y \cap Z (closure under addition)

now we need to show that cu∈Y∩Zcu \in Y \cap Z (closure under scalar multiplication)

Hence Y∩ZY \cap Z is a subspace of XX.


Example: For YY and ZZ are subspaces of XX, show that whether Y∪ZY \cup Z is a subspace of XX or not.
 ~Proof: Prove by contradiction.


Example: Is R2\mathbb{R}^2 a subspace of complex vector space C2\mathbb{C}^2?
 ~Proof: Note that we consider complex vector space, so if x∈R2x \in \mathbb{R}^2 then x∈C2x \in \mathbb{C}^2.

Hence R2\mathbb{R}^2 is not a subspace of C2\mathbb{C}^2.

 ~

Subspaces

Sums of Subspaces

Definition: Suppose W1,...,WmW_1,...,W_m are subspaces of a vector space VV. The sum of W1,...,WmW_1,...,W_m is the set of all possible sums of elements of W1,...,WmW_1,...,W_m:

W1+...+Wm={w1+...+wm:wi∈Wi,i=1,...,m}W_1 + ... + W_m = \{w_1 + ... + w_m : w_i \in W_i, i = 1,...,m\}

Remark: W1+...+WmW_1 + ... + W_m is the smallest subspace of VV containing W1,...,WmW_1,...,W_m.

Direct Sums

Definition: Suppose W1,...,WmW_1,...,W_m are subspaces of a vector space VV. The sum W1+...+WmW_1 + ... + W_m is called a direct sum if each element of W1+...+WmW_1 + ... + W_m can be written in one and only one way as a sum w1+...+wmw_1 + ... + w_m with wi∈Wiw_i \in W_i.

Remark: if W1+...+WmW_1+...+W_m is a direct sum, then W1⊕...⊕WmW_1 \oplus ... \oplus W_m is used to denote the direct sum.

Example: Suppose UjU_j is a subspace of FnF^n of those vectors whose jjth component is zero. As U2={(0,x,...,0)∈Fn:x∈F}U_2 = \{(0,x,..., 0) \in F^n : x \in F\}, Then.

U1⊕...⊕Un=FnU_1 \oplus ... \oplus U_n= F^n

Condition for Direct Sum: Suppose W1,...,WmW_1,...,W_m are subspaces of a vector space VV. Then W1+...+WmW_1 + ... + W_m is a direct sum iff the only way to write 0 as a sum w1+...+wmw_1 + ... + w_m with wi∈Wiw_i \in W_i is by taking w1=...=wm=0w_1 = ... = w_m = 0. In other words, Suppose WW and UU are subspaces of a vector space VV. Then W⊕UW \oplus U is a direct sum iff W∩U={0}W \cap U = \{0\}.

Proof: Proving the statements with iff is equivalent to proving the statements separately, in both directions.

W⊕U is a direct sum  ⟺  W∩U={0}W \oplus U \text{ is a direct sum} \iff W \cap U = \{0\}